Related papers: Galois' Professor's Revenge

We study the height of generators of Galois extensions of the rationals having the alternating group $\mathfrak{A}_n$ as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height…

We establish a quantum Galois correspondence for compact Lie groups of automorphisms acting on a simple vertex operator algebra.

In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of…

In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in $\mathbb Q\left(X_0(N)\right)$ using Hilbert's irreducibility theorem and modular forms. We also…

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer about equality of 2-rank…

Let $K$ be an algebraic number field with ring of integers $\Cal{O}_{K}$, $p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let $\Lambda$ denote the order $\Cal{O}_{K}[G].$ Let $Cl(\Lambda)$ denote the locally free class…

It was suggested by Kontsevich that the Grothendieck-Teichmueller group GT should act on the Duflo isomorphism of su(2) but the corresponding realization of GT turned out to be trivial. We show that a solvable quotient of the motivic Galois…

A self-contained exposition is given of the topological and Galois-theoretic properties of the category of combinatorial 1-complexes, or graphs, very much in the spirit of Stallings. A number of classical, as well as some new results about…

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function…

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups…

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or…

We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific…

Let U be an open subset of a unirational variety. We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U. As a consequence we obtain new proofs of the theorems of Harbater…

This is a survey of our results on the relation between perturbative renormalization and motivic Galois theory. The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose…

We demonstrate how some classical computations of divisor class groups can be obtained using the theory of rational coefficient Weil divisors and related results of Watanabe.

We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with Galois…

Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…

Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of the original problems. This generalizes a…

It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal…

We show that the spectrum of Kontsevich's algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In…